# Some results on second order controllability conditions

## TL;DR

This paper introduces a new, weaker sufficient condition for second order controllability in symmetric systems, accounting for hypersurface geometry and practical switching constraints, and provides constructive control strategies.

## Contribution

It proposes a novel second order controllability condition involving eigenvalues of a symmetric matrix, improving upon classical transversality criteria.

## Key findings

- The new condition is weaker than classical transversality.
- It guarantees controllability with at most one switch.
- Provides explicit control strategies for reaching the target.

## Abstract

For a symmetric system, we want to study the problem of crossing an hypersurface in the neighborhood of a given point, when we suppose that all of the available vector fields are tangent to the hypersurface at the point. Classically one requires transversality of at least one Lie bracket generated by two available vector fields. However such condition does not take into account neither the geometry of the hypersurface nor the practical fact that in order to realize the direction of a Lie bracket one needs three switches among the vector fields in a short time. We find a new sufficient condition that requires a symmetric matrix to have a negative eigenvalue. This sufficient condition, which contains either the case of a transversal Lie bracket and the case of a favorable geometry of the hypersurface, is thus weaker than the classical one and easy to check. Moreover it is constructive since it provides the controls for the vector fields to be used and produces a trajectory with at most one switch to reach the goal.